Lecture 26

Radiation Fields

26.1 Radiation Fields or Far-Field Approximation

Figure 26.1:

In the previous lecture, we have derived the relation of the vector and scalar potentials to the sources J and ρ.

They are given by

A(r) = μ ∫∫∫_V dr′ J(r′) e^(-jβ|r−r′|) / (4π|r − r′|) (26.1.1)

Φ(r) = 1/ε ∫∫∫_V dr′ ρ(r′) e^(-jβ|r−r′|) / (4π|r − r′|) (26.1.2)

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The integrals in (26.1.1) and (26.1.2) are normally untenable, but when the observation point is far from the source, approximation to the integrals can be made giving them a nice physical interpretation.

Figure 26.2:

26.1.1 Far-Field Approximation

When |r| ≫ |r′|, then |r − r′| ≈ r − r′ · r̂, where r = |r| and r′ = |r′|. This approximation can be shown algebraically or by geometrical argument as shown in Figure 26.2. Thus (26.1.1) above becomes

A(r) ≈ μ/4π ∫∫∫_V dr′ μJ(r′) / (r − r′·r̂) e^(-jβr+jβr′·r̂) ≈ μe^(-jβr)/(4πr) ∫∫∫_V dr′ J(r′) e^(jβr′·r̂) (26.1.3)

In the above we have made used of that 1/(1 − Δ) ≈ 1 when Δ is small, but e^(jβΔ) ≠ 1, unless jβΔ ≪ 1. Hence, we keep the exponential term in (26.1.3) but simplify the denominator to arrive at the last expression above.

If we let β = βr̂, and r′ = x̂x′ + ŷy′ + ẑz′, then

e^(jβr′·r̂) = e^(jβ·r′) = e^(jβxx′+jβyy′+jβzz′) (26.1.4)

Therefore (26.1.3) resembles a 3D Fourier transform integral, namely

A(r) ≈ μe^(-jβr)/(4πr) ∫∫∫_V dr′ J(r′) e^(jβ·r′) (26.1.5)

and (26.1.5) can be rewritten as

A(r) ≅ μe^(-jβr)/(4πr) F(β) (26.1.6)

where

F(β) = ∫∫∫_V dr′ J(r′) e^(jβ·r′) (26.1.7)

is the 3D Fourier transform of J(r′) with β = r̂β.

It is to be noted that this is not a normal 3D Fourier transform because |β|^2 = βx^2 + βy^2 + βz^2 = β^2. In other words, the length of the vector β is fixed to be β. In a normal 3D Fourier transform, βx, βy, and βz are independent variables, with values in the range [−∞, ∞], and βx^2 + βy^2 + βz^2 ranges from zero to infinity.

The above is the 3D “Fourier transform” of the current source J(r′) with Fourier variables, βx, βy, βz lying on a sphere of radius β and β = βr̂. This spherical surface in the Fourier space is also called the Ewald’s sphere.

26.1.2 Locally Plane Wave Approximation

We can write r̂ or β in terms of direction cosines in spherical coordinates or that

r̂ = x̂ cos φ sin θ + ŷ sin φ sin θ + ẑ cos θ (26.1.8)

Hence,

F(β) = F(βr̂) = F(β, θ, φ) (26.1.9)

It is not truly a 3D function, since β is fixed. It is a 3D Fourier transform with data restricted on a spherical surface.

Also in (26.1.6), when r ≫ r′·r̂, e^(−jβr) is now a rapidly varying function of r while, F(β) is only a slowly varying function of θ and φ, the observation angles. In other words, the prefactor in (26.1.6), exp(−jβr)/r, can be thought of as resembling a spherical wave. Hence, if one follows a ray of this spherical wave and moves in the r direction, the predominant variation of the field is due to e^(−jβr), whereas the direction of the vector β changes little, and hence F(β) changes little. Furthermore, r′ in (26.1.7) are restricted to small or finite number, making F(β) a weak function of β.

The above shows that in the far field, the wave radiated by a finite source resembles a spherical wave. Moreover, a spherical wave resembles a plane wave when one is sufficiently far from the source. Hence, we can write e^(−jβr) = e^(−jβ·r) where β = r̂β and r = r̂r so that a spherical wave resembles a plane wave locally. This phenomenon is shown in Figure 26.3.

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Figure 26.3: A spherical wave emanating from a source becomes locally a plane wave in the far field.

Then, it is clear that with the plane-wave approximation, ∇ → −jβ = −jβr̂, and

H = 1/μ ∇ × A ≈ −jβ/μ r̂ × (θ̂Aθ + φ̂Aφ) = jβ/μ (θ̂Aφ − φ̂Aθ) (26.1.10)

Similarly,

E = 1/jωε ∇ × H ∼= −jβ/ωε r̂ × H ∼= −jω(θ̂Aθ + φ̂Aφ) (26.1.11)

Notice that β = βr̂ is orthogonal to E and H in the far field, a property of a plane wave. Moreover, there are more than one way to derive the electric field E. Using (26.1.10) for the magnetic field, the electric field can also be written as

E = 1/jωμ ε ∇ × ∇ × A (26.1.12)

Using the formula for the double-curl operator, the above can be rewritten as

E = 1/jωμε (∇∇ · A − ∇²A) = 1/jωμε (−ββ + β²I) · A (26.1.13)

where we have used that ∇²A = −β²A. Alternatively, we can factor β² = ω²με and rewrite the above as

E = −jω (−β̂β̂ + I) · A = −jω (−r̂r̂ + I) · A (26.1.14)

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Since I = r̂r̂ + θ̂θ̂ + φ̂φ̂, then the above becomes

E = −jω (θ̂θ̂ + φ̂φ̂) · A = −jω(θ̂Aθ + φ̂Aφ) (26.1.15)

which is the same as previously derived. It also shows that the electric field is transverse to the β vector. We can also arrive at the above by lettering E = −jωA − ∇Φ, and using the appropriate formula for the scalar potential.

Furthermore, it can be shown that in the far field, using the plane-wave approximation,

|E|/|H| ≈ η (26.1.16)

where η is the intrinsic impedance of free space, which is a property of a plane wave. Moreover, one can show that the time average Poynting’s vector in the far field is

⟨S⟩ ≈ 1/2η |E|^2 r̂ (26.1.17)

which resembles also the property of a plane wave. Since the radiated field is a spherical wave, the Poynting’s vector is radial. Therefore,

⟨S⟩ = r̂Sr(θ, φ) (26.1.18)

The plot of |E(θ, φ)| is termed the far-field pattern or the radiation pattern of an antenna or the source, while the plot of |E(θ, φ)|^2 is its far-field power pattern.

26.1.3 Directive Gain Pattern Revisited

Once the far-field power pattern Sr is known, the total power radiated by the antenna can be found by

PT = ∫₀^π ∫₀^2π r² sin θ dθ dφ Sr(θ, φ) (26.1.19)

The above evaluates to a constant independent of r due to energy conservation. Now assume that this same antenna is radiating isotropically in all directions, then the average power density of this fictitious isotropic radiator as r → ∞ is

Sav = PT / 4πr² (26.1.20)

A dimensionless directive gain pattern can be defined such that

G(θ, φ) = Sr(θ, φ) / Sav = 4πr²Sr(θ, φ) / PT (26.1.21)

The above function is independent of r in the far field since Sr ∼ 1/r² in the far field. As in the Hertzian dipole case, the directivity of an antenna D = max(G(θ, φ)), is the maximum value of the directive gain. It is to be noted that by its mere definition,

∫ dΩ G(θ, φ) = 4π (26.1.22)

266 Electromagnetic Field Theory

where dΩ = ∫₂π₀ ∫_π₀ sin θ dθ dφ. It is seen that since the directive gain pattern is normalized, when the radiation power is directed to the main lobe of the antenna, the corresponding side lobes and back lobes will be diminished.

An antenna also has an effective area or aperture, such that if a plane wave carrying power density denoted by Sinc impinges on the antenna, then the power received by the antenna, Preceived is given by

Preceived = SincAe (26.1.23)

A wonderful relationship exists between the directive gain pattern G(θ, φ) and the effective aperture, namely that1

Ae = λ²/4π G(θ, φ) (26.1.24)

Therefore, the effective aperture of an antenna is also direction dependent.

The directivity and the effective aperture can be enhanced by designing antennas with different gain pattern. When the radiative power of the antenna can be directed to be in a certain direction, then the directive gain and the effective aperture (for that given direction) of the antenna is improved. This is shown in Figure 26.4. Such focussing of the radiation fields of the antenna can be achieved using reflector antennas or array antennas. Array antennas, as shall be shown, work by constructive and destructive wave field of the antenna.

Being able to do point-to-point communications at high data rate is an important modern application of antenna array. Figure 26.5 the gain pattern of a sophisticated antenna array design for 5G applications.

1The proof of this formula is beyond the scope of this course.

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Figure 26.4: The directive gain pattern of an array antenna. The directivity is increased by constructive interference (courtesy of Wikepedia).

268 Electromagnetic Field Theory

Figure 26.5: The directive gain pattern of a sophisticated array antenna for 5G applications (courtesy of Ozeninc.com).

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